transient combustion in mobile gas-permeable propellants

Acta Astronautica, Vol, 3, pp. 573--591. Pergamon Press 1976. Printed in the U.S.A.

Transient combustion in mobile gas-permeable propeilantst~

K. K. KUO,§ J. H. K O O , ¶ T. R. D A V I S II AND G. R. C O A T E S la Department of Mechanical Engineering, The Pennsylvania State University,

University Park, PA 16802, U.S.A.

(Received 10 October 1975; revised 6 February 1976)

Abstract---Gas-permeable propellants possess great potential for producing sizable thrusts within extremely short time intervals. A research program was set up with the objectives: (1) to obtain a deeper understanding of high speed flame propagation and the gas dynamic behavior of two-phase reactive flow systems; and (2) to predict the rates of gasification during transient heterogeneous combustion of granular propellant beds.

The following transient phenomena occur in a few milliseconds: penetration of hot gases into interstitial voids, convective heating of pellets to ignition, granular bed compaction and rapid pressurization from 1 to 2600 atm in a thick-walled steel chamber. Pressure transients and flame-front speeds are measured by high-frequency pressure transducers and ionization probes, respectively. A gaseous pyrogen ignition system utilizing a spark ignition was found most suitable for varying igniter strengths and achieving high reproducibility.

The physico-chemical phenomena described above have been formulated into a theoretical model. The governing equations were derived in the form of coupled, non-linear, inhomogeneous partial differential equations. A stable, fast convergent numerical scheme has been used to solve these hyperbolic partial differential equations.

The results show that the flame front accelerates significantly and the rate of pressurization increases substantially in the downstream direction. The igniter strength and the propellant gasification temperature were found to have pronounced effects on the pressurization process. The theoretical predictions are in close agreement with the experimental data.

Introduction

THIs RESEARCH is stimulated by the potential of gas-permeable propellants for achieving high gasification rates and producing high thrusts during extremely short time intervals.

The objectives of this study are: (1) To advance the state-of-the-art in the knowledge of the combustion

processes in a gas-permeable propellant and/or explosive by formulating a

tPaper presented at the Fifth Colloquium on Gasdynamics of Explosions and Reactive Systems, Bourges, France, 8--11 September 1975.

~tThis paper represents the results of research work performed under Contract DAH CO4-74-G-0016 at the Pennsylvania State University, under the supervision of the Engineering Sciences Division of the U.S. Army Research Office at Research Triangle Park, North Carolina. The support of Dr. A. W. Barrows, Jr. and the technical advice of Mr. N. J. Gerri of the Ballistic Research Laboratories are greatly appreciated. The technical assistance of Mr. T. J. Malmgren ~ of Penn State is also appreciated.

§Assistant Professor. ¶Graduate Assistant in Research. nUndergraduate Assistant in Research.

573

574 K.K. Kuo, J. H. Koo, T. R. Davis and G. R. Coates

complete theoretical model describing the important physical phenomena involved in both gaseous and solid phases.

(2) To solve the theoretical model by selecting a stable, fast convergent, numerical scheme so that the effects of (a) propellant loading density, (b) igniter strength and (c) propellant physico-chemicat characteristics can be studied.

(3) To conduct experimental firings in a thick-walled combustor with a fixed boundary and to explore the early behavior of detonation in condensed, gas-permeable explosives.

The results obtained from this study are directly useful for more accurate predictions of the interior ballistics of artillery systems. The research is also pertinent to the future development of end-burning porous propellant charges to be used in a more predictable and controllable manner for rocket or missile propulsion. In addition, the techniques developed in this investigation can lead toward a better understanding of porous-propellant combustion and the earlier behavior of the overall deflagration-to-detonation transition (DDT).

The phenomena that occur during the transient interval of several milliseconds include the penetration of hot product gases into the granular bed, convective heating of the propellant to ignition, compaction of the granular bed, generation of peak pressure within the chamber, intense pressure pulses at the downstream locations along the granular bed, flame front acceleration, rupture of the shear disc which initially seals the combustion chamber and the depressurization processes after the burst of the shear disc.

Due to the complexity of the physical processes, several different approaches have been proposed. The theoretical methods can be classified into four categories: (1) statistical models (Buyevich 1971; 1972a; 1972b), (2) continuum mechanics models (Krier e t a l . , 1974; 1975), (3) formal averaging models (Gough, 1974a; 1974b) and (4) two-phase fluid dynamic models.t Experimental investigations have also been conducted by various groups.¢ The method employed in this study is the two-phase fluid dynamic model which is developed by formulating the governing equations on the basis that mass, momentum and energy fluxes are balanced over control volumes occupied s e p a r a t e l y by the gas- and particle-phases.

Statistical methods are rejected because in this investigation we are interested in the mean property variations instead of fluctuating quantities. The other reason is that there is not enough information to evaluate the net contribution of the microscopic fluctuation terms without invoking the probability distribution functions and reliable statistical data. The continuum mechanics methods are not considered for the following reasons: (1) the coexistence of solid and gas must be assumed, so that the effect of interpenetration between the two different phases can not be properly described; (2) the derivation of conservation equations for solid and gas phases must come from the splitting of the equations for the mixture; and (3) when steep temperature gradients exist in the solid phase, the energy equation for the solid phase obtained from splitting the mixture energy

tsee Kuo et al. (1973), Kitchens et al. (1973) and Kuo et al. (1974a; 1974b). ~:see Squire et al. (1969), Soper (1973), Gerri et al. (1973) and Alkidas et al. (1975).

Transient combustion in mobile gas-permeable propellants 575

equation simply can not describe the propellant surface temperature and the subsurface gradient. Therefore, the flame spreading rate and heat transfer conditions at the solid-gas interface can not be described adequately. The formal averaging methods are not convenient in the derivation. In that approach, one must initially consider the balance of fluxes from the microscopic point of view. In order to obtain useful solutions, the complicated integrals in the conservation equations have to be simplified by the proper definition of averaged quantities. The final form of the simplified equations should agree with the equations derived from the two-phase fluid dynamic approach. The formal-averaging approach to the two-phase fluid dynamic approach in the derivation of two-phase flow equations is analogous to the relationships between Boltzmann's (Hirschfelder et al., 1954; Williams, 1965) microscopic approach to the Navier-Stokes' (Schlicht- ing, 1968) macroscopic approach in the derivation of single-phase flow equations.

Theoretical model To determine the transient gas dynamic behavior of hot-gas penetration, flame

propagation, chamber pressurization and combustion processes in the granular propellant bed, the mass, momentum and energy equations for the gas phase, and the mass and momentum equations for the solid phase are derived and expressed in a quasi-one-dimensional form. They are approached by considering the balance of fluxes over a control volume small enough to give the desired spatial distributions in the complete system but large enough to contain many solid particles, so that the averaged particle velocity and fractional porosity are meaningful.

The gas-phase control volume is the void portion occupied by the gas in the small elementary volume considered, while the remaining portion occupied by the particles is considered the control volume for the particle phase.

The gas-phase mass equation is

O(d~p) + l O(d~pU, A ) = A,p.rb. (1) Ot A Ox

The particle-phase mass equation is

#[(1 - ~b)pp ] + l O[(1 - ¢ ) p . U p A ] = --Aspprb. (2) Ot A Ox

The gas-phase momentum equation is

O(d?pU,) + 10(~bpU82A)+ g O(d~Pa ) g 0(~'xx~ba) Ot A Ox A Ox A Ox

gP OA = a,rbpp Up - gasD, + - ~ 3-~" (3)

The particle-phase momentum equation is

d[(1-tb)ppUp] 1 a [ ( 1 - ¢ ) p ~ U . 2 A ] g a [ (1 -¢ ) rpA] Ot + -A Ox A Ox

= - A,rbpp Up + gA,D, - g vwp~wp. (4)

576 K. K. Kuo, J. H. Km, T. R. Davis and G. R. Coates

It is important to note that D, is the total drag force between the gas and particle phases. It is equal to the sum of the drag due to the presence of relative velocity between the gas and particle phases and the drag due to the porosity gradient, i.e.

D CD +D f ” P =D --!?b

’ A, ax’

The expression of D, will be discussed in a later part of the theoretical model. The advantage of writing the above two momentum equations in terms of 0, is to facilitate the solution of the hyperbolic problem which is not always totally hyperbolic. The eigenvalues may not be all distinct when the granular bed becomes dispersed. The second advantage is to facilitate the transformation of the governing equations into the characteristic form and to clearly indicate the compatibility relations for the flow properties at the boundaries of the granular bed.

The gas-phase energy equation is

~(~PE)+~~(~PU,EA)+~~(PU,A~) at A ax AJ ax

= - ~w-AsupD~-A,h.(T- TP,)+Aspprb ax J

hchcm+~ 2gJ

+&kh#w p a(u) Q (6) AJ ax ---- AJ at W’

A particle-phase energy equation similar to eqn (6) can be derived in terms of the average bulk temperature of a typical solid particle in the particle-phase control volume considered. However, since the transient interval of the physical process is so short, the temperature profile inside the particles has a very steep gradient. Although the average bulk temperature of a solid particle can be obtained, it would not be useful for determining the ignition condition, the speed of flame front propagation, or the rate of heat transfer from the gas to solid phase. Instead of treating the solid phase energy equation like the mass and momentum equations for the particles, the transient heat conduction equation is considered, so that the propellant surface temperature and the temperature gradient can be accurately obtained.

Following the motion of a given particle, the heat equation written in the Lagrangian time derivative and spherical coordinate of the particle is

(7)

(8)

The initial condition for eqn (7) is:

t = 0: T,(O, r) = To.

Transient combustion in mobile gas -permeable propellants 577

The boundary condition before the thermal wave has penetrated to the center of the sphere are:

r = rpo:

aT, r = O: -~ - (t, O) = 0

aTe. h , ( t ) (t, r~) = ~ [T( t ) - Tp= (t)]

(9)

(10)

where the total heat transfer coefficient, h,, is the sum of the convection and radiation coefficients; i.e.

.~_ 2 h,( t)=- h c ( t ) + 6 , ~ [ T ( t ) + To:(t)][T2(t) Tp,(t)] (11)

where the gas emissivity is taken to be unity. The heat-up equation and its initial and boundary conditions are recasted by

the integral method in the same manner as given by Kuo and Summerfield (1974b) to yield a first order ordinary differential equation describing the increase of propellant surface temperature with respect to time, i.e.

[DT , , ] = 8r~ k, t ] / , / D, i. r6, - + h,81

L r~o ~J (12)

where 6 is the thermal wave penetration depth in a spherical particle, measured from the particle surface; 6 as a function of time is given by:

~(t) = 3r~o[T,~(t)- To] (13)

[ T , ~ ( t ) - To] r~oh,(t) [ T ( t ) - Tps(t)] kp

Similarly, an ordinary differential equation describing the changes of Tp, after the thermal wave has penetrated the full radius of a spherical particle is obtained. Since the transient interval of the physical process is only a few milliseconds, the particle surface reaches the ignition condition long before the thermal wave penetrates to the center of the particle; the equation describing the changes of Tp= after the thermal wave penetration is not given here. After Tps is solved, a simplified ignition temperature criterion is used to determine the burning condition of particles along the propellant bed. In order to avoid the step function changes in burning rate, a two-temperature criterion is used to achieve full ignition within a finite time interval; when Tp= is equal to Tab~, the particle starts to gasify and as soon as T~ reaches T,~. the full ignition condition is reached.

In addition to the above governing equations, the equations of state for gas- and particle-phases must be specified. The co-volume effect becomes important at high pressures, so the Noble-Abel dense gas law, also known as the Clausius


Equation (Strehlow, 1968),

is used as the equation of state for the gas phase. The statement of a constant density for the solid-propellant particles serves as the equation of state for the particles.

To complete the theoretical model it is necessary to specify several empirical correlations: the intragranular stress transmitted through the packed granular particles, the flow resistance due to the drag force between particle and gas phases, the convective heat transfer coefficient and the regression rates of the solid propellant particles. The constitutive law (Koo, 1975) used for the intragranular normal stress calculations is:

=

ppC 2 ch (1 ~)) g( l_~b)cb ( t h c - t h ) - P ( 1 ~_ if th~b~

- P if 4, >~,,

(15)

where the is the critical porosity above which there is no direct contact between particles, and C is the speed of sound transmitted in granular material.

From the fluidization condition of the gas-particle system in the granular bed, the bed can be divided into a non-fluidized (packed) region and a fluidized (dispersed) region. Since Ergun's (1952) equation is valid only for 1< (Rep/(1-40)<4000 and 0.4< ~b <0.65, it has very limited application to the combustion of granular propellants under high convective burning situations. A series of cold-flow resistance measurements were made by using a cylindrical chamber of the same inner diameter as the combustion chamber and the same type of propellant, but under non-fluidized non-combustion conditions. The correlation obtained (Kuo et al., 1976) is

D. # ( U , - Up)(1 - ~b) { ( Rep ,~o.~,[ = ~g-~p~ 276.23 + 5.05 \~--Z-~] j (16)

which is valid for l<(Repl(1-~b))<24,000. For the fluidized region, the expression for Do is deduced from Andersson's (1961) expression which is valid for porosities ranging from 0.45 to 1.0 and Rep from 0.003 to 2000. Although the Reynolds number range is not wide enough to cover the variation in the overall transient process, it is the best correlation available in the literature. Since the values of Dv in the fluidized region are substantially smaller than those of the nonfluidized conditions, any inaccuracy introduced by extrapolating the drag correlation for a fluidized bed to a higher Reynolds number will not have a significant influence on the calculated results. The drag correlation for the nonfluidized region, however, is very important in the momentum equations.

For convective heat transfer calculations, Denton's (1951) formula is used for

Transient combustion in mobile gas-permeable propellants 579

the non-ttuidized region, hence

k [plU~ - Up .~r lo.7 hc 0.65 __ I'g . ] prU3.

r. [ I-~ J (17)

It is valid for Re . from 500 to 50,000 and porosity around 0.37. Although the porosity range is very narrow for this correlation, it represents the porosity of most of the packed, unburned portion of the granular bed in combustion experiments. For tiuidized regions, hc is obtained from Rowe and Claxton's (1965) correlation,

rp 1 - ( 1 - ~b) m+ PrmRepm (18)

where the exponent of the Reynolds number is expressed as

0.6667 + 1.55 Re . -°z8 m = 4.65 Re.-°'2s+ 1 (19)

The Rowe--Claxton correlation is valid for 10 < Re . < 10 7. For burning rate calculations, the Lenoir and Robbillard (1957) semiempirical

burning rate law is adopted:

/3rbp. rb = aP" + Kehc exp - p l U g _ U. I} (20)

where he, the local zero-blowing convective heat-transfer coefficient, is obtained from eqn (17) or eqn (18). The erosive-burning constant, K., and the erosive-burning exponent,/3, used in this investigation are listed in Table 1.

Before the necessary boundary conditions are considered, the governing equations for the gas and particle phases are simplified further after an order of magnitude analysis. The higher-order terms neglected are: (a) the viscous normal stress in the gas-phase momentum equation, (b) the shear force at the combustor wall for the particles in the particle momentum equation (this is justified since the contact surface area between the particles and chamber wall is small and also the initial porosity considered is high), (c) the gas-phase heat conduction term, (d) the work done by the viscous normal stress in the gas-phase energy equation, (e) the heat loss 0w to the chamber wall in the extremely short transient combustion experiments, (f) the rate of pressure work for the dilatation of the gaseous control volume in the gas-phase energy equation, and (g) the rate of change of the total heat transfer coefficient in the calculation of propellant surface temperature.

The system of governing equations, after the above simplifications, becomes a set of six first order, coupled, non-linear, inhomogeneous partial differential equations. It is found in the study that the eigenvalues of the system are six distinct real numbers under non-fluidized conditions. The characteristic directions

580 K . K . K u o , J . H . K o o , T . R . Davis and G . R . Coates

Table 1. Data used in the numerical computations

pp = 1.6 g / c m ~

ap = 0 .945 x 10 -3 c m Z / s e c

kp = 5 .30 x 10 4 c a l / c m . s e c _ O K

k = 0 .265 x 1 0 - ' c a l / c m - s e c - ° K

/z = 0 . 44615 × 10 -~ g / c m - s e c

b = 1.26 c m 3 / g

y ( 5 0 0 ° K ) = 1.4

y ( 3 0 0 0 ° K ) = 1.26

Pr = 0.7

rpo -- 0 . 41275 x 10 -~ c m

~c = 0.3995 A = 0 .4745 c m 2

CL = 15.24 c m To = 2 9 3 . 9 1 ° K

Po = 1033.23 g / c m 2

d,o = 0 .3995

K , = 0.1 c m 3 - ° K / c a l

f l = 105

e,, = 1.0

T.b, = 5 6 0 ° K

T, , . = 6 3 0 ° K Th~ = 2 5 0 0 ° K

TI = 283 I ° K

Ca (rpo) = O. 1969

rp ( u n d e t e r r e d ) = 0 . 0 3 2 9 4 5 c m

a ~ = 0 .562 × l O - ~ ( c m / s e c ) / ( g / c m 2 ) ° -~7

a3o = 0 .2814 × lO-*(cm[sec)](g/cm:) . . . . c ~ : o Cd~ 0 = 0 . 3 0

noo = 0 .8867

n3o = 0 .8020

a a n d n a r e linearly interpolated from aoo, a3o and noo, n3o according to the values o f C,~ at the instantaneous s u r f a c e o f the propellant grain.

are described by:

dx) = U, + C,, ( ) (dx) d x = U , - in

dx = U p - C, dt- vl

The subscripts I, II and III represent the right-running, left-running, and gaseous-path characteristic lines in the gas phase. The subscripts IV, V and VI represent the right-running, left-running and particle-path characteristic lines in the solid phase.

When the granular bed becomes fluidized, the speed of sound is not transmitted through the dispersed particles; therefore, C is equal to zero. However, the speed of sound in the gas phase is never zero; hence, eqn (21) remains the same but eqn (22) changes to:

-- (23)

when 4} >4}c. Due to this reduction of six characteristic lines into four characteristic lines when the granular bed changes from non-ttuidized to fluidized conditions, the governing equations are not totally hyperbolic. The solution of these equations becomes quite complicated. The boundary conditions must be well specified.


The total number of boundary conditions required depends upon the flow conditions at the igniter end of the granular bed. (As far as the current calculation is concerned, the chamber is sealed by the shear disc at the downstream end; the phenomena after the rupture of the shear disc are beyond the scope of this paper.) To specify conditions at the igniter end, a separate control volume is considered, which is located immediately at the entrance of the granular bed. When this entrance control volume contains both gas and particles, six ordinary differential equations are used together with the necessary compatibility relations to determine the average flow properties in the entrance volume and also the properties at the boundary between the entrance volume and the granular bed. The compatibility relations are derived directly by transforming the governing equations into their characteristic forms. When the entrance section contains gas only, three ordinary differential equations describing the time rate of change of flow properties of the gas phase are solved together with the appropriate compatibility relations for the boundary parameters needed. The total number of compatibility relations used in the calculation depends upon the flow directions of the gases and particles at the boundary and also the fluidization conditions. The boundary conditions at the shear disc end are relatively simple; before the shear disc is ruptured, the gas and particle velocities are all zero.

The initial conditions required to solve the system of equations are the initial distributions of gas temperature, pressure, velocity, particle velocity, propellant surface temperature and fractional porosity.

Numerical solution The modified two-step Richtmyer (1962) scheme incorporated with predictor-

corrector calculations was used to solve the governing partial differential equations. In using the predictor-corrector technique, the non-linear terms are treated in an explicit manner.

For interior points, the finite difference diffusing scheme (Lax, 1954) using central differences in space and forward differences in time is employed for the first-step calculations. The difference algorithm for the predictor and corrector calculations is given below.

Predictor:

OU= .[ U{ +IA -- 0-5(U{+1 'Jr" Ui--1)] at At

a_u = (u {+ , - u~_,). Ox 2Ax '

(24)

Corrector:

o U = [Ui j+ l - 0.5(Uj+ 1 + Uj_I) ] at At

aU [ ( 1 - 0 ) ( U { + , u { _ , ) + j+,A , + , , _ _ = - 0 ( u , + , - U , _ , )] Ox 2Ax

(25)


The modified Leap-Frog scheme (Forsythe et al., 1967) using central differences in both space and time is employed for the second-step calculations. The Crank-Nicolson parameter (Fox, 1962), 0, is used to help stabilize the numerical solution. The difference algorithm for the second-step in the predictor and corrector calculations is given below.

Predictor:

oU__ (mi j+2a- mi j ) o__U= ( Ui+ll÷' -- Ui__l )l+l Ot 2At Ox 2Ax ' (26)

C orrector:

OU ( U, '+~- Ul ) Ot 2At

aU [(I O)(U~4- I U{--I) ~- O( I+2~ J+2~' j+l ,+l - - = -- -- mi+l -- Ui - I ) "~(Ui - - l - - U, I)] Ox 4Ax

(27)

The non-linear coefficients and inhomogeneous terms were treated explicitly in the above predictor-corrector technique. The boundary points are calculated by using physical and compatability relations to achieve a high degree of numerical stability. The compatability relations used can also be regarded as the extraneous boundary conditions (Kuo, 1971) which are needed when the central difference technique is used for space derivatives in the hyperbolic partial differential equations. A small amount of artificial viscosity (Ames, 1969) is introduced in the calculation to filter out the high frequency components. The important data used in the numerical computations are tabulated in Table 1.

Experimental apparatus Experiments to verify the theoretical model are carried out using the

cylindrical chamber shown in Fig. 1. The spherical propellant used for the tests is type WC-870,t with an average particle diameter of 0.8255 mm. The particles are packed in a chamber 15.25 cm long with 0.778 cm i.d. to form a propellant bed with an average weight of 6.95 g.

To measure pressure and flame front speed, ultra-high frequency Minihat pressure gages (Brosseau, 1970) and ionization probes [Dynasen CA-1040] (Bernecker et al., 1974) are equally spaced along the chamber. The Minihat pressure gases are statically calibrated by a dead weight tester up to 6600 atm at an accuracy of 0.1% at Ballistic Research Laboratories of Aberdeen Proving Ground. In order to maximize chamber strength and also to verify the one-dimensional gas dynamic behavior assumed in the theoretical model, the transducers and ionization probes are placed spirally along the chamber. The downstream boundary of the chamber is a 0.81 mm stainless steel shear disc (burst diaphragm) which provides a fixed boundary and also serves as a safety valve.

tWC-870 granular propellant is manufactured by Olin Corp., Conn. The maior constituents are 79.28% of NC (13.15% N2), 9.97% NG and 5.2% DBP.


" 15.24cm , r 12.38

Multi-perforated J" 9.21 ~ 1 Gauge 4 nozzle ~-6D3-'J | / -pressure

12B6 | l transducer Hz- ~ GII I -'~ G 7 ~ GI4/~GG , Sheardisk7

I t t . ' I ' o l 2 I S~pork lPropella:t~/~lonizatlon dSih~ or / plug bed ~ pins retainer L--ignifio n UCombustion

chamber chamber

Fig. 1. Schematic drawing of the ignition and combustion chamber.

A gaseous pyrogen ignition system is used to ignite the propellant bed. This type of ignition system was chosen over the conventional impact ignition primer because it is capable of achieving both high reproducibility as well as varying igniter strength and duration. These features are not characteristic of pellet igniters (Gerri et al., 1973).

The igniter strength and reproducibility are controlled by regulating the gaseous mixture pressure and the fuel-oxidant ratio. The reactant combination of gaseous hydrogen and oxygen was chosen for the igniter because of its wide flammability limits and because it is easily ignited by sparks. They are well mixed in the ignition chamber before spark ignition. Two spark plugs are used to insure a successful ignition in every firing.

The ignition and combustion chambers are separated by a multi-perforated convergent nozzle. The upstream side of the nozzle is covered and tightly sealed by a tape to prevent unburned reactant gases from entering the propellant bed before ignition. The design of the multi-perforated nozzle is intended specifically to provide a radially uniform heating of the granular propellant bed, to achieve a one-dimensional flow condition in the axial direction, and also to provide a one-dimensional compaction of the bed.

The transient data obtained from pressure transducers and ionization pins are recorded on a multi-channel F.M. tape recorder at a speed of 60i.p.s. in order to provide totally time-correlated results. The analog data are then recorded on a transient waveform digitizer (Biomation Waveform Recorder Model 1015) which obtains 100 data points per millisecond of transient data. The data is expanded by three orders of magnitude in time and plotted on a conventional x -y plotter at a rate of one data point per second.

D i s c u s s i o n o | r e s u l t s

The data obtained from pressure transducers show that the rate of pressurization invariably increases in the downstream direction. A typical set of pressure-time traces for the five gauges used is shown in Fig. 2. Gauge 1 (G 1) was positioned in the igniter chamber, and because of a choked backflow condition at


S

PmoxOfGI = I , ~ 0 kg/crn 2 G2=2,270

Time, msec

Fig. 2. Measured pressure-time traces from a typical firing (traces are offset vertically for clarity).

the nozzle, it was found to lag behind the rise in pressure at Gauge 2 location. The four other gauges, G2 to G5, were positioned along the propellant bed with an equal spacing of 3.175 cm.

The pressure traces measured in the granular bed indicate that an upstream gauge (like G2) senses the pressurization much sooner than the gauges at downstream positions. However, the rate of pressurization is significantly higher for the downstream gauges. It is also interesting to note that the pressure peak for G5 is slightly ahead of G2. This is because the pressurization of G5 is cut off sooner due to the effect of rarefaction waves caused by the rupture of the shear disc. When the depressurization effect due to shear disc rupture overcomes the pressurization due to propellant gasification at a given station, the pressure-time traces start to decline. The pressure at all stations drops back to one atmosphere, ending a transient interval of about 5 msec.

Figure 3 and 4 show a comparison of the theoretical predictions with the composite pressure time traces for G2 and G5 respectively for a series of six firings under the same experimental conditions. The shaded region represents the degree of scatter of the experimental data. The solid lines represent the theoretical predictions for the rise in pressure at the corresponding stations. Since the theoretical calculation is limited to the time interval prior to the rupture of the shear disc, no comparison is made for the depressurization processes. It is shown clearly from the above comparison that the predicted pressure-time variations at these two different gauge locations not only have the right magnitude but also have reasonable slopes. The pressure rise times are also not far from the

Transient combustion in mobile -gas permeable propellants 585

3000.

N 2OOO E

G.. iooo

Rupture of :~:

. $ ~ "

' Theoretical prediction

i~7!:~! Range of experimental data

:.:'.

.~';: ::~:

'::.'...~'.:

I 2 3 4 5 Time, msec

Fig. 3. Comparison of theoretically predicted G2 pressure trace with the composite pressure traces of six experimental firings..

3000

Theoretical prediction

~.!~i?i.~ii Range of experimental clata

Rupture of N E 2000 shear disc--~,

Q. I000

0 I 2 3 4 Time, msec

Fig. 4. Comparison of theoretically predicted G5 pressure trace with the composite pressure traces of six experimental firings.

experimental measurements. The comparison of theoretical prediction of the pressure rises at G3 and G4 with the experimental data need not be shown here; they also compare quite closely with experimental results. The calculated results reveal the same trend as shown in Fig. 2 that the pressure at a downstream location begins to rise at a later time than the upstream pressure. The calculated results also show that the pressure rise becomes steeper in the downstream directions which is in agreement with experimental observations.

586 K . K . Kuo, J. H. Koo, T. R. Davis and G. R. Coates

In Fig. 5 the calculated flame spreading rate is compared with test data from three separate firings with ionization pin measurements. (Three other firings of the same series consisting of the six runs mentioned above had no flame front measurements.) The calculated results again check quite closely with the experimental data. Both the theoretical results and experimental data indicate a significant increase in flame velocity as the combustion wave propagates in the shear disc direction.

Figure 6 shows pressure distributions calculated for various times, with the locus of the ignition front (dashed curve) superimposed. Beginning with the trace (t = 1.26msec), the pressure distribution quickly develops a hump and the gradient on its right side becomes steeper with increasing time from the increased gasification rate in the combustion zone. The pressure gradient grows to the left of

2.0

1.5

E

0; I O E k--

- - Theoretical prediction

o,5- E] Firing No. 40 / ~ Firing No. 38 ( ~ Firing No. 39

o ~' Distance, cm

Fig. 5. Comparison of theoretically predicted flame front position with experimental data.

5000 Pressure distribution

. . . . Locus of flame front

.~ 200o

35

,ooo y/ \ \ /

o.0 5D io.o 15o

Distance, cm

Fig. 6. Calculated pressure distributions at various times.


the peak due to the reverse flow of gases from the granular bed to the igniter chamber. Thus, when the pressure at the head end of the granular bed becomes higher than that in the igniter chamber, the igniter chamber serves as a mass and energy sink to the combustion chamber. The resulting reverse flow of gases also makes the peak more pronounced. Further, the reverse gas flow eventually causes the pressure trace of t = 1.54 to cross that of t = 1.42 msec. (This phenomenon produces the dip on the rising portion of the pressure-time traces at G2 location shown in Fig. 3.) Near the end of the chamber the locus of the ignition front curves sharply upward (Fig. 6), due to the extremely rapid pressurization near the shear disc.

Figure 7 shows the temperature distributions (at times corresponding to Fig. 6), with the locus of the ignition front shown by a dashed curve. The advancement of the combustion wave in the granular bed is readily seen. Early in the transient, the pressure in the system is low, so in order to heat the granular propellants to ignition condition, the gas temperature at the ignition front must be high. As the pressure rises, the heat transfer from the gas to the unburned propellant becomes more efficient and the ignition front locus descends. Finally, near the end of the granular bed, where the gas velocity is small, the convective heat transfer effect decreases considerably and, therefore, the gas temperature at the ignition front again rises.

Figure 8 shows the corresponding gas velocity distributions. Early in the transient, igniter gas flows into the granular bed and the gas velocity is greatest near the nozzle end. Later, with combustion occurring in the granular bed, a peak develops in the velocity profile driven by the pressure gradient near the ignition front. Still later, the pressure gradient in the granular bed near the nozzle causes reverse flow of hot gases into the igniter chamber; therefore, the gas velocity is negative in the left portion of the combustion chamber.

In Fig. 9 the particle velocity distributions are shown on the same scale to emphasize the fact that particle velocities are in general much lower than the gas

4000 Temperature distribution

. . . . Locus of flame front

~ o. IOOO- 4 2 ~

0 I I 0.0 5.0 I0.0 15.0 Distance, cm

Fig. 7. Calculated gas temperature distributions at various times.


200

I00

E

°

o --I00 •

19 ~ 0.42 ~ 1 . 6 5 5

-- 2O0 I I i 0.0 50 I0.0 15.0

Distance, crn

Fig. 8, Ca lcula ted gas veloci ty distr ibut ions at var ious times.

200

#. ¢: ~oo

o

- I00

-200

t = 1.54 m s e c

0 . 4 2 ~ 5

0.98 1.26 142

I I ! 0.0 50 too 150

Distance, cm

Fig. 9. Calculated particle velocity distributions at various times.

velocity. The motion of the particles are generated by the pressure gradients developed in the granular bed. Later in the transient, a fraction of the particles move in the reverse direction near the nozzle end. Again, since the inertia of the particles is much greater than that of gas molecules, their reverse speed is much lower than that of the gas at the nozzle end.

C o n d u s i o n s (1) A gaseous igniter system has been developed successfully in producing

ignitions in granular propellant beds. (2) Transient wave phenomena and flame spreading are measured by high

frequency pressure gages and ionization pins. Experimental results indicate that: (a) the rate of pressurization increases in the downstream direction and the

pressure peak developed in the granular bed travels downstream;


(b) t he f lame s p r e a d i n g r a t e i n c r e a s e s s igni f icant ly in the d o w n s t r e a m d i r ec t i on ; (c) the f lame f ron t p r o p a g a t i o n is n e a r l y one d i m e n s i o n a l , i n d e p e n d e n t o f the

r ad ia l p o s i t i o n o f the i on i za t i on p ins ; (d) a s t r o n g e r igni te r c a u s e s f a s t e r f lame sp read ing and a l so a h igher r a t e o f

p r e s s u r i z a t i o n . (3) A s u c c e s s f u l t h e o r e t i c a l m o d e l has b e e n d e v e l o p e d to d e s c r i b e the

c o m b u s t i o n of m o b i l e g r a n u l a r p rope l l an t s . A s table , f a s t - c o n v e r g e n t n u m e r i c a l s c h e m e is u s e d fo r so lv ing the c o m p l e t e s y s t e m of equa t ions . T h e a b o v e e x p e r i m e n t a l l y o b s e r v e d p h e n o m e n a w e r e a l so d i s p l a y e d in t he so lu t ions o f t he

t h e o r e t i c a l mode l . (4) T h e c a l c u l a t e d p r e s s u r e - t i m e t r ace s and f lame s p r e a d i n g r a t e s ag ree

c l o s e l y wi th the e x p e r i m e n t a l da ta .

References Alkidas, A., Morris, S. O., Caveny, L. H. and Summerfield, M. (1975) An experimental study of

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Ames, W. F. (1969) Numerical Methods for Partial Differential Equations, pp. 234-238. Barnes & Noble, New York.

Andersson, K. E. B. (1961) Pressure drop in ideal fluidization, Chem. Engng Sci. 15, 276-297. Bernecker, R. R. and Price, D. (1974) Studies in transition from deflagration to detonation in granular

explosives--I. Experimental arrangement and behavior of explosives which fail to exhibit detonation, Comb. and Flame 22(I), 111-117.

Brosseau, T. L. (1970) Development of the Minihat Pressure Transducer For Use in the Extreme Environments of Small Caliber Gun Barrels, BRL MR 2072, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland.

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Ergun, S. (1952) Fluid flow through packed columns, Chem. Engng Progr. 48(2), 89-94. Forsythe, G. E. and Wasow, W. R. (1967) Finite Difference Methods for Partial Differential Equations,

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74-1, Naval Ordnance Station, Indian Head, Maryland. Gough, P. S. (1974b) The flow of a compressible gas through an aggregate of mobile reacting particles,

Ph.D. Thesis, Mech. Engng Dept., McGill University, Montreal, Quebec, Canada. Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B. (1954) Molecular Theory of Gases and Liquids. Wiley,

New York. Kitchens, C. W., Jr. and Gerri, N. J. (1973) Numerical and experimental investigation of flame

spreading and gas flow in gun propellants, Presented in JANNAF Safety Meeting on Combustion. Koo, J. H. (1975) Theoretical modeling and numerical solution of transient combustion processes in

mobile granular propellant beds, M.S. Thesis, Mech. Engng Dept., The Pennsylvania State University, University Park, Pennsylvania.


Krier, H.. Van Tassell, W., Rajan, S. and VerShaw, J. T. (1974) Model of Gun Propellant Flame Spreading and Combustion, BRL CR 147, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland.

Krier, H. and Rajan, S. (1975) Flame spreading and combustion in packed beds of propellant grains, AIAA 13th Aerospace Sciences Meeting, Pasadena, Calif., Paper No. 75-240.

Kuo, K. K. (1971) Theory of flame front propagation in porous propellant charges under confinement, Ph.D. Thesis, AMS RN 1000-T, pp. 73-80. Princeton University, Princeton, N.J.

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Kuo, K. K. and Summerfield, M. (1974b) High speed combustion of mobile granular solid propellants: wave structure and the equivalent Rankine-Hugoniot relation, In Fi[teenth (International) Symposium on Combustion, pp. 515-527. The Combustion Institute, Pittsburgh, Pennsylvania.

Kuo, K. K. and Nydegger, C. C. (1976) Cold Flow Resistance Measurement and Correlation in a Packed Bed o[ WC-870 Spherical Propellants, to be published.

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Lenoir, J. M. and Robillard, G. (1957) A mathematical method to predict the effects of erosive burning in solid-propellant rockets, In Sixth Symposium (International) on Combustion, pp. 663--667. Reinhold, New York.

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Appendix A

Nomenclature E total stored energy (internal plus a pre-exponential factor for burning kinetic energy) per unit mass

rate g gravitational constant A cross-sectional area of the cylindri- h, the average convection heat-transfer

cal combustion chamber coefficient over pellets A~ specific surface area of the granular hch,= enthaipy of the propellant gas at

propellants, the surface exposed flame temperature to fluid per unit volume h, the total heat-transfer coefficient, the

C speed of sound in particle phase unit-surface conductance Cd deterrant concentration ] heat to work conversion factor CL combustion chamber length k thermal conductivity of gases Dp drag force due to porosity gradient kp thermal conductivity of particles D, the total drag force between the gas K, erosive burning constant

and particle phases, D, = Do + Do n pressure exponent for the burning Do drag force acting on gases by the rate law

particles per unit wetted area of P pressure particles Pr Prandtl number

(D/Dt), Lagrangian time derivative when q rate of conduction heat transfer per observer follows the motion of a unit area particle, (O/St) + Up(OlOx) Q~ rate of heat loss to the chamber wall


per unit spatial volume of gas- particle system

r radial distance from the center of a spherical particle

rb burning rate of the solid propellant rp radius of pellets R gas constant

Rep Reynolds number, 2r~p4~ [ U~ - Up I[/~ t time

T temperature Tab~ ablation temperature of the solid

propellant T~ adiabatic flame temperature of pel-

lets Th, temperature of the hot igniter gas T~,. ignition temperature Tp, particle surface temperature To initial temperature of pellets U velocity with respect to laboratory

coordinates x distance from the beginning of the

granular propellant bed thermal diffusivity

/3 erosive burning exponent in the Lenoir-Robbillard burning-rate law

3' specific heat ratio ,~ thermal wave penetration depth in a

spherical particle, measured from the particle surface

ep emissivity of the particle /~ dynamic viscosity of the gas phase p gas density

pp density of pellets ~wp wetted perimeter between the

chamber wall and particle phase ~-p normal stress transmitted in solid

propellants ~'wp shear stress between the chamber

wall and particle phase r~ viscous normal stress in gas phase

fractional porosity ~c critical porosity

0 Crank-Nicolson parameter tr Stefan-Boltzmann constant

Subscripts g gas phase p particle phase 0 initial state

00 0% deterrant concentration 30 30% deterrant concentration

Superscript A predicted values in numerical

scheme

transient combustion in mobile gas-permeable propellants

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